Moving a vertex makes only half of the connected faces move in the right way
Your problem comes down to changing the internal triangulation done by Blender, as described by Robin Betts here: flipping internal triangulation
In your particular case: select problematic faces, Ctrl + T to triangulate, change Quad Method to Fixed Alternate, select the new edges and dissolve them:

If you want to make a script automatically fixing such quads, this could be a starting point: Select concave quads